Showing papers on "Lyapunov function published in 1978"

Stability criteria for large-scale systems

Abstract: Recent research into large-scale system stability has proceeded via two apparently unrelated approaches. For Lyapunov stability, it is assumed that the system can be broken down into a number of subsystems, and that for each subsystem one can find a Lyapunov function (or something akin to a Lyapunov function). The alternative approach is an input-output approach; stability criteria are derived by assuming that each subsystem has finite gain. The input-output method has also been applied to interconnections of passive and of conic subsystems. This paper attempts to unify many of the previous results, by studying linear interconnections of so-called "dissipative" subsystems. A single matrix condition is given which ensures both input-output stability and Lyapunov stability. The result is then specialized to cover interconnections of some special types of dissipative systems, namely finite gain systems, passive systems, and conic systems.

Stability of multidimensional scalar and matrix polynomials

Abstract: A comprehensive study of multidimensional stability and related problems of scalar and matrix polynomials is presented in this survey paper. In particular, applications of this study to stability of multidimensional recursive digital and continuous filters, to synthesis of network with commensurate and noncommensurate transmission lines, and to numerical stability of stiff differential equations are enumerated. A novel approach to the multidimensional stability study is the classification of various regions of analyticity. Various computational tests for checking these regions are presented. These include the classical ones based on inners and symmetric matrix approach, table form, local positivity, Lyapunov test, the impulse response tests, the cepstral method and the graphical ones based on Nyquist-like tests. A thorough discussion and comparison of the computational complexities which arise in the various tests are included. A critical view of the progress made during the last two decades on multidimensional stability is presented in the conclusions. The latter also includes some research topics for future investigations. An extensive list of references constitutes a major part of this survey.

Control of Dynamical Systems.

Abstract: : The problem of arbitrarily assigning closed loop poles of a linear multivariable system developed a new method, a generalization of the classical root locus method. Studies have also been conducted on the attainment to stable solutions of model matching problems. A technique was developed, based on a modified minimum energy regulator problem, to obtain feedback stabilization of linear time varying differential systems. Two methods of parameter identification for linear differential systems were developed. A study was made of bilinear control systems with applications to parachute gliding systems and the pursuit-evasion missile control problem. Studies were made of linear operator feedback for the compensation and control of multivariable systems. A number of computational methods and techniques for control problems with diffusion models were developed, in addition to the study of the application of Monte Carlo methods for the optimization of constrained noisy systems. The study of bifurcation problems has been pursued from the abstract viewpoint and for specific applications. Studies were continued for systems described by ordinary and functional differential equations.

Positive diagonal solutions to the Lyapunov equations

Abstract: We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.

On norm bounds for algebraic Riccati and Lyapunov equations

Abstract: New lower bounds on the spectral norms of the positive definite solutions to the continuos and discrete algebraic matrix Riccati and Lyapunov equations are derived. These bounds are much easier to compute than previously available bounds and appear to be considerably tighter in many cases.

Lyapunov stability of interconnected systems: Decomposition into strongly connected subsystems

Abstract: New Lyapnnov results for uniform asymptotic stability, exponential stability, Instability, complete instability, and uniform ultimate boundedness of solutions for a class of interconnected systems described by nonlinear time-varying ordinary differential equations are established. The present results make use of graph theoretic decomposition techniques to transform large-scale systems into an interconnection of strongly connected components (subsystems), and they also make use of the properties of stability preserving mappings. The analysis of the overall interconnected system is then accomplished in terms of the qualitative properties of the subsystems (strongly connected components) and the stability preserving properties of the system interconnections. A typical result is of the following form. If every subsystem (strongly connected component) is uniformly asymptotically stable and if all system interconnections are stability preserving, then the overall interconnected system is uniformly asymptotically stable. To demonstrate the applicability of these results to physical systems, a specific example is considered. The principal advantages of the present method over existing Lyapunov results for interconnected systems are as follows. 1) The present results make it possible to address the "largeness" of complex systems, since the identification of the strongly connected components (subsystems) can be accomplished by means of efficient computer algorithms. 2) The present stability results are applicable to interconnected systems with unstable subsystems (i.e., the original nontransformed system description may involve unstable subsystems). The main disadvantages of the present method over many existing Lyapunov results are the following. 1) The analysis is not accomplished in terms of the original system structure. 2) When a system consists of only one strongly connected component, the present method cannot be used to advantage, while other methods may.

Multimachine power systems: Stability, decomposition, and aggregation

Abstract: A new approach is proposed for transient stability analysis of interconnected power systems, which is based upon the concept of vector Lyapunov functions and the decomposition-aggregation method. The approach results in an exact procedure for computation of stability region estimates which are expressed explicitly in terms of system parameters. More refined models of the subsystems can be readily accommodated by the new approach. In particular, the transfer conductances are included in the present study, a feature which is almost exclusively missing in transient stability analysis of multimachine systems by Lyapunov's method.

Asymptotic Stability of a Class of Non-linear Singularly Perturbed Systems†

Abstract: Sufficient conditions are obtained to guarantee the asymptotic stability of a class of non-linear singularly perturbed systems. A procedure for constructing a Lyapunov function for such a class of systems is given, and a clearly defined domain of attraction of the equilibrium is obtained. A stabilizing feedback control for such systems is also proposed.

On some algorithms for design of optimal constrained regulators

Abstract: Two algorithms for computation of optimal feedback gains for output constrained regulators are considered. One algorithm consists of an iterative solution of Lyapunov equations. It is shown that it does not always converge even for good initial values. The other algorithm includes a solution of a nonlinear equation in each iteration. This algorithm is superior to the first one for considered examples.

The absolute stability of high-order discrete-time systems utilizing the saturation nonlinearity

Abstract: Previously, we investigated the stability of a class of discrete-time filters involving a restricted class of linear systems, the all-pole systems, and a single, specific, memoryless nonlinearity-the saturation nonlinearity. A rather effective criterion was derived for determining if a given system is free of all periodic oscillations except the trivial null solution. The derivation relied on the observation that certain rather exclusive properties, called passivity properties, are associated with the saturation nonlinearity. Here, we generalize the results in two directions. First, the linear part of the system is allowed to be quite general, i.e., both poles and zeros may occur in the transfer function. Thus in the context of digital filters, the restriction to the direct-form realization is lifted. Secondly, the criterion obtained here guarantees the absolute stability of the system and not just the absence of nontrivial periodic oscillations. Thus the possibility of aperiodic or "chaotic" behavior is eliminated. The broadened formulation enhances the utility of the criterion, making it applicable to the large class of recursive digital processing systems, including digital filters, which use the saturation arithmetic to cope with overflow in the course of forming the sum of the results of many multiplicative operations. Also, the methods developed indicate an approach for the stability investigation of large digital transmission networks containing nonlinearities. The technique used for deriving the new result is entirely different. The new element is the demonstration that the hypothesis which is in the form of a condition in the frequency domain implies the existence of a Lyapunov function.

On stability theory

Abstract: It is found that under mild assumptions, feedback system stability can be concluded if one can 'topologically separate' the infinite-dimensional function space containing the system's dynamical input-output relations into two regions, one region containing the dynamical input-output relation of the 'feedforward' element of the system and the other region containing the dynamical output-input relation of the 'feedback' element. Nonlinear system stability criteria of both the input-output type and the state-space (Lyapunov) type are interpreted in this context. The abstract generality and conceptual simplicity afforded by the topological separation perspective clarifies some of the basic issues underlying stability theory and serves to suggest improvements in existing stability criteria. A generalization of Zames' conic-relation stability criterion is proved, laying the foundation for improved multivariable generalizations of the frequency-domain circle stability criterion for nonlinear systems.

The application of the Poincaré-transform to the Lotka-Volterra model

Abstract: In this article, we show how to apply the Poincare-transform to the general Lotka-Volterra model and investigate the question of the global asymptotic stability of the locally stable equilibrium point. An attempt is made to estimate the domain of attraction when the critical point is not globally stable. Results for the two dimensional case, due to Goh [3], are improved.

Control Logic to Track Outputs of a Command Generator

Abstract: A procedure is presented for synthesizing time-invariant control logic to cause the outputs of a linear plant to track the outputs of an unforced (or randomly forced) linear dynamic system. The control logic uses feedforward of the reference system state variables and feedback of the plant state variables. The feedforward gains are obtained from the solution of a linear algebraic matrix equation of the Lyapunov type. The feedback gains are the usual regulator gains, determined to stabilize (or augment the stability of) the plant, possibly including integral control. The method is applied here to the design of control logic for: 1) a second-order servomechanism to follow a linearly increasing (ramp) signal, 2) an unstable third-order system with two controls to track two separate ramp signals, and 3) a sixth-order system with two controls to track a constant signal and an exponentially decreasing signal (aircraft landing-flare or glide-slope-capture with constant velocity).

Stability of General Systems in Biological, Physical and Social Sciences

Abstract: The systems of nonlinear differential equations represent the mathematical model of several general systems in biological, physical and social sciences [1] One of the most versatile techniques in the theory of nonlinear differential systems is the second method of Lyapunov [1,2] The notion of vector Lyapunov functions, together with the theory of systems of differential inequalities, provides a very general comparison principle by means of which a number of qualitative properties of solutions of deterministic functional differential systems are studied in a unified way [3] It is natural to expect such an extension to stochastic functional differential systems The obtained comparison principles to stochastic functional differential systems have been utilized to study the stability behavior of hereditary stochastic general systems

Stability results for multiple volterra integral equations

Abstract: Lyapunov stability, uniform stability, and asymp­ totic stability for a class of systems of multiple Volterra integral equations are studied. Stability results are ob­ tained using a representation for the solution in terms of a fundamental solution (a generalization of the fundamental matrix in the theory of ordinary differential equations). Criteria for stability in terms of the fundamental solution are established for the general linear equation under consideration. A nonlinear perturbed equation is studied and results concerning the preservation of stabili­ ties from the linear to the perturbed equation are given. Lipschitz and little o type nonlinearities are considered. The general results are then applied to several special equations and conditions for various stabilities are given in terms of the kernels in these equations. The results established may also be used to study stability of the characteristic value problem for hyperbolic partial differential equations.

Time lag versus stability

Abstract: By employing Lyapunov functions and the theory of differential inequalities in the context of a suitable minimal class of functions, sufficient conditions are given for stability of functional differential systems. It is shown that the dominant diagonal property of a matrix provides a suitable mechanism to resolve "time-lag vs. stability" problem.

Stability theory for countably infinite systems of differential equations

Abstract: New stability results for a class of countably infinite systems of differential equations are established. We consider those systems which may be viewed as an interconnection of countably infinitely many free or isolated subsystems. Throughout, the analysis is accomplished in terms of simpler subsystems and in terms of the system interconnecting structure. This approach makes it often possible to circumvent difficulties usually encountered in the application of the Lyapunov approach to complex systems with intricate structure. Both scalar Lyapunov functions and vector Lyapunov functions are used in the analysis. The applicability of the present results is demonstrated by means of several motivating examples, including a neural model.

Comparison Theorems and Instability with Applications to Numerical Integration, Satellite Orbits and Averaging

Abstract: Using converse theorems of Lyapunov, and a theorem of Babuska which relates numerical integration and stability under persistent disturbances, we obtain upper bounds for the error in numerically computed satellite orbits. These estimates are applied to give some insight into the effect that various stabilization techniques of Baumgarte and Stiefel have on the numerical error, and the relation between stability and error estimates in the method of averaging.

Stability of Large-Scale Nonlinear Systems- Quadratic-Order Theory of Composite-System Method Using M-Matrices

Abstract: Abshact--’Ibe composite-system metbod for nnalyzing sfabii of large-scale systems is stndied focming on the quaddc+rder tkmeans using M-matriw. Here, by tbe term “amposite-system method”, we refer tothemethodtodecompasealarge-scalesystemintosmallersubsystems and to make two-step analysis (Le.., fii to analyze subsystems and second to eambine the results to rednce the property of the *le). ’llmries about Lyapunov stabii and about input-ontput stability are hi from a unified standpii and their mutual relation is clarXed. As an application, simple frequency domain stabiity aiteria are given for certain rlasses of multi-input mnlti-output system. The contents are generally useful for stabfity analysis of large-seale nonlinear system.

Improved adaptive observer with optimal convergence

Abstract: A simple technique for simultaneous parameter identification and state estimation of linear discrete systems is presented. The gradient method is used for identification and the global stability of the scheme is guaranteed by the use of Lyapunov's method. The optimal adaptive gains are also considered for more rapid convergence.

A comparsion of lyapunov and hyperstability approaches to adaptive control

Abstract: : The aim of this brief paper is to examine the conditions which have to be satisfied for the two approaches to be successfully applied to adaptive observers and controllers and also to demonstrate that they are entirely equivalent. Using a typical error model it is shown that hyperstability and asymptotic hyperstability are achieved under exactly the same conditions as stability and asymptotic stability in the sense of Lyapunov. In those cases where the adaptive control problem remains unresolved, the difficulties encountered using Lyapunov's method are shown to have their counterparts in hyperstability theory as well.